Either your paragraph style has the strike through feature activated or you have a character style selected with strike through attribute. If nothing is selected click on none in Character Styles, then select the text and click also on none.

Unfortunately it's not a strikethrough. I think it's a paragraph style override that I would like to hide, but I cannot find the place where I can toggle it off. Show/hide paragraph style overrides - or something of the sort?

Both the blue highlights and the red strikethroughs can indicate local overrides to text formatting, specifically manually applied tracking/kerning. Try removing all overrides and see if they disappear.

They do disappear if I remove the overrides. Thing is, I would like to keep the overrides, but not see them highlighted. Any way I can do that? I went into Preferences > Composition and deselected absolutely everything - they're still there...

I would strongly recommend not to work with any overrides because these are often problematic in a later workflow and cause most of final production issues. Create a proper style and use that. This will make your work more effective.

They do disappear if I remove the overrides. Thing is, I would like to keep the overrides, but not see them highlighted. Any way I can do that? I went into Preferences > Composition and deselected absolutely everything - they're still there...

Everything is still there even after turning off everything in the preferences. Also, just noticed some highlighting doesn't disappear when removing overrides and the highlighted text has no changes made to kerning or tracking. Some text is highlighted in purple, some in brown, some in green.

Yes, I absolutely agree with you on that - as a general rule. Problem is this specific document has to go on as it is - I just thought there would be an easy way to toggle the highlighting I posted above on or off...

The blue highlight should go away when you change your prefs. But since it didn't (and since the red strikethrough shouldn't be visible), try running the Show Text Overrides script from In-Tools. It should toggle that stuff off. InDesign Scripts | in-tools.com